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Published for SISSA by Springer
Received: November 5, 2013 Accepted: December 16, 2013 Published: January 28, 2014
Minimal Liouville gravity correlation numbers from
Douglas string equation
Alexander Belavin,a,b,c Boris Dubrovind,e,f and Baur Mukhametzhanova,g
aL.D.Landau Institute for Theoretical Physics,
prospect academica Semenova 1a, 142432 Chernogolovka, Russia
bMoscow Institute of Physics and Technology,
Insitutsky pereulok 9, 141700 Dolgoprudny, Russia
cInstitute for Information Transmission Problems,
Bol’shoy karetni pereulok 19, 127994, Moscow, Russia
dInternational School of Advanced Studies (SISSA),
Via Bonomea 265, 34136 Trieste, Italy
eN.N. Bogolyubov Laboratory for Geometrical Methods in Mathematical Physics,
Moscow State University “M.V.Lomonosov”, Leninskie Gory 1, 119899 Moscow, Russia
fV.A. Steklov Mathematical Institute,
Gubkina street 8, 119991 Moscow, Russia
gDepartment of Physics, Harvard University,
17 Oxford street, 02138 Cambridge, U.S.A.
E-mail: belavin@itp.ac.ru,dubrovin@sissa.it,baur@itp.ac.ru
Abstract:We continue the study of (q, p) Minimal Liouville Gravity with the help of Dou-glas string equation. We generalize the results of [1,2], where Lee-Yang series (2, 2s + 1) was studied, to (3, 3s + p0) Minimal Liouville Gravity, where p0 = 1, 2. We demonstrate
that there exist such coordinates τm,n on the space of the perturbed Minimal Liouville
Gravity theories, in which the partition function of the theory is determined by the Dou-glas string equation. The coordinates τm,nare related in a non-linear fashion to the natural
coupling constants λm,n of the perturbations of Minimal Lioville Gravity by the physical
operators Om,n. We find this relation from the requirement that the correlation numbers
in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature [3–5].
Keywords: 2D Gravity, Conformal and W Symmetry, Matrix Models ArXiv ePrint: 1310.5659
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Contents
1 Introduction 2
2 Minimal Liouville gravity 4
2.1 Minimal Models 4
2.2 Coupling to Liouville theory 6
2.3 Contact terms 7
2.4 Three- and four-point correlation numbers 8
3 Douglas string equation approach to Minimal Gravity 9
3.1 Douglas string equation 9
3.2 Free energy in the Douglas approach 12
4 (q, p) = (2, 2s + 1) Minimal Gravity 14
4.1 Dimensionless setup 16
4.2 One- and two-point correlation numbers 16
4.3 Three-point correlation numbers 17
4.4 Four-point correlation numbers 18
5 (q, p) = (3, 3s + p0) Minimal Gravity 19
5.1 Resonance relations 20
5.2 Solutions of the string equations 21
5.3 Picking out the contour 22
5.4 Dimensional analysis 23
5.5 Zero-, one- and two-point numbers 24
5.6 Three-point correlation numbers 25
5.7 Four-point correlation numbers 29
6 Conclusion 32
A Frobenius manifolds 32
A.1 Basic definitions. Deformed flat coordinates on a Frobenius manifold 33
A.2 An example: 2D TFT of the An type and Frobenius structure on the space
of polynomials of degree n + 1 37
A.3 From Frobenius manifolds to integrable hierarchies 40
A.4 Frobenius manifold of Antype and dispersionless limit of the Gelfand-Dickey
hierarchy 44
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1 Introduction
Since the invention of the Minimal Liouville Gravity [6, 7], one of its important problems was to explicitly find arbitrary n-point correlation functions of physical operators in this theory. The Minimal Liouville Gravity represents probably the simplest example of two-dimensional quantum gravity, when the matter sector is taken to be a (q, p) Minimal Model of CFT [8]. However even in this theory it turned out to be a highly non-trivial problem to calculate n-point correlation functions. The main difficulty is to calculate the integrals over the moduli space of Riemann surfaces with n punctures. Some progress in this direction was achieved in [5], where, using higher Liouville equations of motion [9], a new technique for this sort of integrals was developed by Alexei Zamolodchikov and one of the authors. In this way the 4-point functions on the sphere in Minimal Gravity were calculated there [5]. The most natural physical objects to calculate in Minimal Liouville Gravity are the n-point correlation functions
Zm1n1...mNnN = hOm1n1. . . OmNnNi, Om,n = Z
MO
m,n (1.1)
where Om,n — are some local physical observables which are to be introduced in the
sec-tion 2. We also restrict ourselves to the case when the manifold M has the topology of the sphere. Often the quantities Zm1n1...mNnN are referred to as correlation numbers. It is convenient to consider the generating function
ZL(λ) = hexp X m,n λm,nOm,ni, (1.2) Zm1n1...mNnN = ∂ ∂λm1n1 . . . ∂ ∂λmNnN λ=0 ZL(λ) (1.3)
The generating function (1.2) can be understood as the partition function of the perturbed theory. For that reason we call it partition function. We can think of the coupling constants λm,nas of the coordinates on the space of the perturbed Minimal Liouville Gravity theories.
On the other hand since the late 1980s another, discrete, approach [10–16] to 2-dimensional gravity (as opposed to the continuous approach of Minimal Liouville Gravity) has been developing.1 In this approach a fluctuating 2-dimensional surface is approximated by an ensemble of graphs. The continuous geometry is restored in the scaling limit, when large size graphs dominate. This approach is realized through Matrix Models (find a survey of these ideas in [17,18]).
Since both approaches are based on the same idea of 2-dimensional fluctuating geom-etry, they were expected to give the same results for physical quantities. This was checked by calculations in some particular models [3,7,19].
In 1989 Douglas in his seminal paper [20] had shown that the partition function in the discrete approach satisfies a differential equation and the so-called “string equation”, which is usually referred to as Douglas string equation. The times τm,n in the generalized KdV
hi-erarchy play a role of the perturbation parameters λm,n in Minimal Gravity. Indeed, it was
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shown in [20] that the scaling dimensions of the times τm,n coincide with those of the
cou-pling constants λm,n in Minimal Gravity. However a na¨ıve identification of the times τm,n
and coupling constants λm,n leads to inconsistencies. Namely, in Minimal Gravity one has
conformal and fusion selection rules. For instance, the one-point correlation numbers of all operators, except the unity operator, must be zero, the two-point correlation numbers must be diagonal and there are similar restrictions to higher point correlation numbers. These conformal and fusion selection rules are not satisfied in the Douglas approach if we na¨ıvely identify τm,n and λm,n. This problem was first pointed out by Moore, Seiberg and
Stau-dacher in [1]. There they also proposed how this problem can be solved. The idea of [1] was that due to possible contact terms in the correlators, the times τm,nin the Douglas approach
and coupling constants λm,nin Minimal Liouville Gravity are related in a non-linear fashion
τm,n = λm,n+
X
m1n1m2n2
Cm1n1m2n2
m,n λm1n1λm2n2 + . . . (1.4)
Appropriate choice of this substitution allowed them to explicitly establish the correspon-dence up to two-point correlation numbers in (2, 2s + 1) Minimal Gravity.
After the 3- and 4-point correlation numbers in Minimal Gravity had been calculated in [3–5], it became possible to make more explicit checks against Douglas equation ap-proach. These kind of checks were performed in [2] for (2, 2s + 1) Minimal Gravity, where, using the ideas of [1], the full correspondence between Douglas equation approach and Minimal Gravity was proposed for arbitrary n-point correlation numbers.
The aim of this paper is to generalize the results of [1, 2] to the case of (3, 3s + p0)
Minimal Gravity.2 Our analysis is based on the following assumptions:
• There exist special coordinates τm,n in the space of perturbed Minimal Liouville
Gravities such that the partition function of Minimal Liouville Gravity satisfies, as in matrix models, a partial differential equation and the Douglas string equation. • Besides, one can show that the solution of the Douglas string equation satisfies the
generalized KdV hierarchy equations [21,22].
• The times τm,n are related to the natural coordinates λm,n (1.2) on the space of the
perturbed Minimal Liouville Gravities by a non-linear transformation like in (1.4). We establish the form of this transformation τ (λ) by the requirement that the cor-relation numbers (1.3), which are the coefficients of the expansion of the partition function in the coordinates λm,n, satisfy the conformal and fusion selection rules.
• The differential equation for the partition function and the Douglas string equation define the partition function as logarithm of Sato’s tau-function [24] of the dispersion-less generalized KdV hierarchy [21,22] with initial conditions defined by the Douglas string equation.
To simplify our analysis we need an explicit expression for the partition function. We find such an expression using the relation of the Douglas string equation and dispersionless
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KdV hierarchy with the Frobenius manifold structure [22, 25]. This relation leads to a convenient integral representation of the partition function (3.30) for all (q, p). Using this expression we analyse the case (3, 3s + p0) where p0 = 1, 2. As in [1] and [2] the relation
be-tween Minimal Gravity and Douglas equation approach is found to be non-trivial because of the contact terms in the correlation functions. Namely, the scaling dimensions of the coupling constants λm,n in (1.2) coincide with those of the times τm,n in the Douglas
equa-tion approach. However, the relaequa-tion between them is found to be non-linear. This relaequa-tion is established by the requirement that there exists a polynomial τm,n({λm,n}) such that
after the substitution τm,n → τm,n({λm,n}) the partition function in the Douglas equation
approach coincides with the partition function (1.2) in Minimal Gravity and the obtained correlation numbers satisfy the conformal and fusion selection rules. In this way the rela-tion τ (λ) is implicitly established in terms of the Jacobi polynomials Pn(a,b)up to the third
powers of λ in (1.4). Also, after establishing the relation τ (λ), we evaluate the correlation numbers when they are non-zero. The results are in agreement with the direct evaluations of three-point [3,4] and four-point [5] correlation numbers in Liouville Minimal Gravity.
In spite of the agreement in the non-zero three- and four-point correlation numbers, using the substitution τ (λ) in the partition function in the Douglas equation approach we managed to satisfy only a part of the conformal and fusion selection rules, while the other part remains unsatisfied. We will discuss this issue in the final parts of the paper.
In section 2 we review the conformal and fusion selection rules and the results for three-and four-point correlation numbers in Minimal Gravity. Also we discuss in this section the subtlety of contact terms which makes the relation τ (λ) non-trivial. Then in section 3 we discuss the Douglas approach to Minimal Gravity. In section 4 we warm up by rederiving the results of the paper [2] for (2, 2s + 1) Minimal Gravity in a slightly different way. And finally, we make generalization to the case (3, 3s + p0) in section 5. The concluding remarks
and discussion of the results can be found in section 6. To make the text self-consistent we give a review of ideas on Frobenius manifolds in appendix A.
2 Minimal Liouville gravity
The Liouville Gravity consists of Liouville theory of a scalar φ and some conformal matter sector. One of the simplest cases is when the matter sector is taken to be a (q, p) Minimal Model of CFT [8]. This theory is called Minimal Liouville Gravity.
2.1 Minimal Models
The Minimal Models of CFT Mq,p, labelled by two co-prime integers (q, p), have a finite
number of primary fields, which are enumerated by the Kac table: Φm,n, where m =
1, . . . , q − 1 and n = 1, . . . , p − 1. Due to the relation
Φm,n = Φq−m,p−n (2.1)
only a half of the fields Φm,n are independent.
In this paper we consider the Minimal Models M2,2s+1 and M3,3s+p0, where p0= 1, 2. In these models it suffices to consider the fields from the first raw Φ1,k. Let us introduce a
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notation
Φk= Φ1,k+1. (2.2)
In the model (2, 2s + 1) the independent fields are Φk, 0 ≤ k ≤ s − 1. In the model
(3, 3s + p0) the independent fields are Φk, 0 ≤ k ≤ p − 2.
The operator product expansion (OPE) for these fields is
[Φk1][Φk2] =
I(k1,k2) X
k=|k1−k2|:2
[Φk] (2.3)
where, as usually, [Φk] denotes the contribution of the irreducible Virasoro representation
with the highest state Φk. Here the symbol k = |k1− k2| : 2 denotes summation that goes
from k = |k1− k2| till I(k1, k2) with the step 2 where
I(k1, k2) = min(k1+ k2, 2p − k1− k2− 4). (2.4)
The small conformal group of fractional-linear transformations together with OPE (2.3) puts strong constraints on correlation functions. Particularly, many of them are zero. For instance the small conformal group constraints one- and two- point correlation functions
hΦk(x)i = 0, k 6= 0, (2.5)
hΦk1(x1)Φk2(x2)i = 0, k1 6= k2. (2.6)
For higher correlation numbers we use the OPE to bring the n-point correlation func-tion to a combinafunc-tion of two-point funcfunc-tions and then use the rules (2.5), (2.6). For instance the three-point correlation functions satisfy
hΦk1Φk2Φk3i = 0, k3> I(k1, k2) = (
k1+ k2, k1+ k2≤ p − 2
2p − k1− k2− 4, k1+ k2 > p − 2
(2.7)
where we assume that k1 ≤ k2 ≤ k3. For four-point correlation functions we have
hΦk1Φk2Φk3Φk4i = 0, k4> I(I(k1, k2), k3) (2.8) where we assume that k1 ≤ k2 ≤ k3≤ k4 and
I(I(k1, k2), k3) = k1+ k2+ k3; k1+ k2≤ p − 2, k1+ k2+ k3 ≤ p − 2 2(p − 2) − k1− k2+ k3; k1+ k2 > p − 2, k1+ k2− k3≥ p − 2 2(p − 2) − k1− k2− k3; k1+ k2 ≤ p − 2, k1+ k2+ k3> p − 2 k1+ k2− k3; k1+ k2> p − 2, k1+ k2− k3 < p − 2 (2.9)
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2.2 Coupling to Liouville theory
In this section we consider the interaction of Minimal Models, considered in the previous section, with the Liouville filed theory. The Minimal Model plays the role of conformal matter. For this reason the resulting theory is called Minimal Liouville Gravity.
The Polyakov’s continuous approach to two-dimensional quantum gravity [6] is defined through the path integral over two-dimensional Riemannian metrics gµν interacting with
some conformal matter. In conformal gauge gµν = eφˆgµν it leads to Liouville action
SL= 1 4π Z M pˆggˆµν∂µφ∂νφ + Q ˆRφ + 4πµe2bφ d2x (2.10)
where ˆgµν is some fixed background metric, µ — cosmological constant and parameters
Q, b are related to the central charge cL of the Liouville theory
cL= 1 + 6Q2, Q = b + b−1. (2.11)
The central charge cM of the conformal matter is related to the central charge of the
Liouville theory by the Weyl anomaly cancellation condition
cL+ cM = 26. (2.12)
In the case of (q, p) Minimal Liouville Gravity where the conformal matter is (q, p) Minimal Model of CFT, we have b =qqp.
The observables of the (q, p) Minimal Liouville Gravity are constructed as cohomologies of an appropriate BRST operator. They are enumerated by the same integers as primary fields in corresponding Minimal Model and denoted as Om,n. Explicitly they have the form
Om,n =
Z
x∈MO
m,n(x), Om,n(x) = Φm,n(x)e2bδm,nφ(x)pˆgd2x (2.13)
where Φm,n — primary fields of the (q, p) CFT Minimal Model, considered in the previous
section. The numbers δm.n are the so-called gravitational dimensions [7]
δm,n = p + q − |pm − qn|
2q . (2.14)
The operators Om,n have the scaling property
Om,n ∼ µ−δm,n. (2.15)
Obviously, the operators Om,n satisfy the same selection rules (2.5) – (2.8) as the primary
fields Φm,n do.
The most easily defined physical object in Minimal Liouville Gravity is correlation numbers
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where the average is taken over the fluctuations of the metric and of the conformal mat-ter fields. Again it is convenient to consider the generating function of these correlation numbers ZL(λ) = * expX m,n λm,nOm,n + . (2.17)
We can also think about the generating function as the partition function of the Minimal Gravity perturbed by the operators Om,nwith the coupling constants λm,n. For that reason
the generating function will often be called partition function.
This kind of generating functions and correlation numbers will be the main object of our study. In particular we will check that the results of [5] for the correlation numbers in (q, p) Minimal Gravity agree with the those obtained from the so-called Douglas string equation, which will be introduced in the subsequent sections.
2.3 Contact terms
In this section we discuss one important subtlety which makes the definition of the correla-tion numbers ambiguous. The correlacorrela-tion numbers (2.16) involve integration over n points on the two-dimensional surface M
Zm1n1...mNnN = Z
hOm1,n1(x1) . . . OmN,nN(xN)id
2x
1. . . d2xN (2.18)
Let us notice that there could be contact delta-like terms when two or more points xi
are coincident. Such terms are not controlled by the conformal field theory and thus make the definition of correlation numbers ambiguous. For instance, in CFT calculations, like those in [5], one just disregards all the contact terms and supposes that the conformal se-lection rules are the same for operators Om,n and Om,n. On the other hand in the discrete
approach, and in stemming from it Douglas string equation, one might expect that the contact terms are treated differently and conformal selection rules for correlators of Om,n
are not satisfied. This is what actually happens. Apparently this phenomenon was first observed in [1].
In other words the ambiguity in contact terms leads to the fact that we can add to the n-point correlation numbers some k-point correlation numbers, where k < n. For instance, we can add a one-point correlation number to the two-point correlation number
hOm1,n1Om2,n2i → hOm1,n1Om2,n2i + X
m,n
A(m1n1)(m2n2)
m,n hOm,ni. (2.19)
It is easy to see that such a substitution is equivalent to a change of coupling constants in the generating function (2.17)
λm,n→ λm,n+
X
m1,n1,m2,n2
A(m1n1)(m2n2)
m,n λm1n1λm2n2. (2.20)
Also we must include here possible changes of the background [1]. Namely, when λm,n is
proportional to an integer power of the cosmological constant µ = λ1,1, we have
λm,n → Aµδm,n+ λm,n+
X
m1,n1,m2,n2
A(m1n1)(m2n2)
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Generalizing this result we arrive at a convenient way of describing the ambiguity in contact terms. Namely, any addition of contact terms is equivalent to some non-linear polynomial change of coupling constants in the generating function (2.17)
λm,n → Cm,nµδm,n+ λm,n+ X m1,n1 X m2,n2 C(m1n1)(m2n2) m,n λm1n1λm2n2+ + X m1,n1 X m2,n2 X m3,n3 C(m1n1)(m2n2)(m3n3) m,n λm1n1λm2n2λm3n3 + . . . . (2.22)
So, there can be many different “systems of coordinates” of λm,n. In Minimal Gravity we
use one “system of coordinates” and the Douglas string equation uses another one. In Min-imal Gravity there is a strong restriction on the substitutions of coupling constants (2.22). Indeed, the observables Om,n have certain mass dimension (2.15). Consequently the
cou-pling constants also have certain mass dimension
λm,n ∼ µδm,n (2.23)
This restricts the possible non-linear terms in (2.22). Namely, all the acceptable terms in (2.22) must have the same dimension as λm,n
δm,n= δm1,n1+ δm2,n2+ δm3,n3+ . . . (2.24) We usually call (2.24) the resonance condition and substitutions of coupling constants (2.22) resonance relations.
In the Liouville Minimal Gravity the resonance conditions (2.24) turn out to be very common. So we must resolve this ambiguity somehow. We do it in the following way. We evaluate the free energy F(t) from the Douglas equation. Then make a change of variables t = t(λ) like in (2.22) and require that after this substitution the conformal selection rules are satisfied.
2.4 Three- and four-point correlation numbers
In [3, 4] the three-point correlation numbers were calculated. The four-point correlation numbers were calculated in [5]. We will compare these results with those from the Douglas string equation.
A priori, the normalizations of the operators Om,n and of the correlators are not
co-incident in Minimal Gravity and in the Douglas string equation. Thus to make sensible comparisons we write down the quantities which do not depend on the normalizations of operators and correlators
hhOm1,n1Om2,n2Om3,n3ii 2 Q3 i=1hhO2mi,niii = Q3 i=1|mip− niq| p(p + q)(p − q) (2.25) hhOm1,n1Om2,n2Om3,n3Om4,n4ii Q4 i=1hhO2mi,niii 12 = (2.26) = Q4 i=1|mip− niq| 1 2 2p(p + q)(p − q) 4 X i=2 m1−1 X r=−(m1−1) n1−1 X t=−(n1−1) |(mi− r)p − (ni− t)q| − m1n1(m1p+ n1q)
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Additionally it needs to be mentioned that the four-point correlation numbers (2.26) were obtained under certain assumptions. In the particular case when mi = 1, i = 1, . . . , 4
the assumption looks as follows (cf. [5])
n1+ n4 ≤ n2+ n3. (2.27)
(suppose n1≤ · · · ≤ n4).
3 Douglas string equation approach to Minimal Gravity
3.1 Douglas string equation
Due to Douglas [20] the free energy of the matrix model, corresponding to the partition function (2.17) of (q, p) Minimal Gravity, is described in the following way. One takes differential operators P and Q of the form
ˆ Q = dq+ q−1 X α=1 uα(x)dq−α−1, (3.1) ˆ P = 1 q 1 +p q ˆ Qpq + q−1 X α=1 X k=1 1 q k + α q tk,αQˆk+ α q−1 ! + (3.2)
where d = dxd, the pseudo-differential operators ˆQaq = da+ . . . are understood as series in d and (. . . )+means that only non-negative powers of d are taken in the series expansion over
d. The constants tk,αare usually called “times”. This terminology comes from the
connec-tion with KdV-type integrable hierarchies (see appendix A below; the constants tk,α
coin-cide, up to a numerical factor, with the time variables tα
k of the Gelfand-Dickey hierarchy).
Then the so-called string equation is defined as
[ ˆP , ˆQ] = 1. (3.3)
The string equation yields a system of differential equations on uα(x). The free energy
F(˜t) then satisfies the equation
∂2F
∂x2 = u ∗
1 (3.4)
where u∗
α is an appropriate solution of the string equation (3.3).
The first term in (3.2) describes the critical point of the matrix model and corresponds to the Minimal Liouville Gravity. Other terms in (3.2) correspond to the Minimal Liouville Gravity perturbed by primary operators. In (3.2) we did not indicate the interval in which the index k is varied. We will do this later from the condition that the perturbations by the times tk,α correspond to the coupling constants λm,n of the perturbation of the Minimal
Liouville Gravity by all primary operators from (q, p) minimal CFT.
In the present work we are interested in calculating the correlation numbers on the sphere. In the language of Douglas string equation this means that we are interested in the
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quasiclassical (dispersionless) limit of these equations. In this limit the “momentum oper-ator” dxd is replaced with a variable3 p and the commutator in string equation is replaced with Poisson bracket
[P, Q] = 1 → {P, Q} =∂P∂x∂Q∂p − ∂P∂p ∂Q∂x = 1, (3.5) Q = pq+ q−1 X α=1 uα(x)pq−α−1, (3.6) P = 1 q 1 +p q Qpq + q−1 X α=1 X k=1 1 q k + α q tk,αQk+ α q−1 ! + (3.7)
where P and Q are the symbols of the differential operators ˆP and ˆQ. Qaq is understood as series expansion in p and (. . . )+ means that only non-negative powers of p are taken in
this expansion.
The string equation (3.3) is actually a system of differential equations on the functions uα. Heuristically, taking the quasiclassical limit means that we must through out all the
derivative terms except the first derivatives (leave terms linear in duα
dx and get rid of terms
like d2uα
dx2 ,d 3u
α
dx3 , . . . ).
It was shown in [26] that either in the full form (3.3) or the quasiclassical limit (3.5), one can take the first integral of the string equation. In the quasiclassical limit after this integration the Douglas string equation (3.3) becomes [26] (see the proof in appendix A)
∂S ∂uα(x) = 0 (3.8) where S[uα(x)] = Ss+1,p0 + q−1 X α=1 X k=0 tk,αSk,α (3.9)
where (q, p) = (q, sq + p0), 1 ≤ p0 ≤ q − 1, t0,1 = x and we introduced a notation
Sk,α= Res(Qk+
α
q) (3.10)
Here Q is the polynomial (3.6) and the residue is taken at p = ∞. The details can be
found in the appendix A.4
The times t0,α in (3.9) with 2 ≤ α ≤ q − 1, which do not appear in (3.2), are the
integration constants. The time t0,1, which does not appear in (3.2) either, comes from the
integration of the unit on the right hand side in (3.3).
For obvious reasons, the equation (3.8) is usually referred to as the principle of least action. Since it is equivalent to the Douglas string equation, we will be using the former for convenience. For instance, as was mentioned above, we are interested in the quasiclassical limit of the string equation. For the string equation itself it means that we are left with
3Not to be confused with p in the (q, p) minimal CFT. 4The functions S
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the system of the first order differential equations. On the other hand for the principle of least action equation (3.8) it means that in the quasiclassical limit we are left with the system of algebraic equations. So it is convenient for us to use the principle of least action instead of the Douglas string equation.
Now let us make more accurate statement about the perturbations we need to be added in (3.9). Namely, we ask in what region we should vary the index k in (3.9) in order to make the correspondence with the perturbation by all primary operators in Minimal Liou-ville Gravity. This question can be answered with the help of dimensional analysis. The construction of the Douglas string equation happens to have a scaling properties similar to those of the Minimal Liouville Gravity. Historically, this coincidence of the “gravitational dimensions” was actually the first reason to believe that two descriptions describe the same phenomena.
To identify the free energy of the matrix model with the partition function of the Liou-ville Minimal Gravity on the sphere, we first analyse the dimensions. The (q, p) LiouLiou-ville Minimal Gravity partition function on the sphere has the following scaling property [7]
ZL∼ µ
p+q
q (3.11)
where µ is the cosmological constant.
On the other hand the equation (3.4) gives that
Z ∼ x2u1 ∼ p2(p+q) (3.12)
where the scaling dimension of x is determined from the equation {P, Q} = 1 and scaling dimension of uα is determined from the constraint that all terms in the polynomial Q are
of the same order
x ∼ pp+q−1, uα ∼ pα+1. (3.13)
Thus if we want F ∼ ZL we have
p ∼ µ2q1 . (3.14)
This determines
tk,α∼ µ
s+1−k
2 +p0−α2q . (3.15)
In particular ts−1,p0 ∼ µ. We want to identify these times and their dimensions with the dimensions of the coupling constants λm,n (2.23) or, equivalently, with the dimensions of
the observables Om,n in the Minimal Liouville Gravity (2.15).
In the (q, p) Minimal Gravity one has the following scaling dimensions of the coupling constants λm,n (1 ≤ m ≤ q − 1; 1 ≤ n ≤ p − 1)(2.14), see eqs. (2.23) above
δm,n = p + q − |pm − qn|
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Let us find among the times tk,α those which correspond to the coupling
con-stant λm,n in Minimal Gravity. To do this observe that the expression for the
“ac-tion” (3.9) determines the dimensions of the times tk,α. Let us write it in the form
S = ResQp+qq +P
m,nτm,nQam,n
, where we changed enumeration of times (k, α) to some new enumeration (m, n). Then the dimension of times τm,n is (since Q ∼ pq ∼ µ
1 2) τm,n∼ µ 1 2 p+q q −am,n . (3.17)
Now we see that in order for the dimension of the time τm,n to coincide with (3.16),
we need to put am,n = |pm−qn|q . Thus we have another expression for the action, in which
the correspondence with the operators from Minimal Gravity is clear S = Res Qp+qq + q−1 X m=1 p−1 X n=1 τm,nQ |pm−qn| q ! . (3.18)
The relation between two types of enumeration (k, α) and (m, n) can be established from the equation
|pm − qn|
q = k +
α
q. (3.19)
For instance µ ∼ ts−1,p0 = τ1,1, where (q, p) = (q, sq + p0), 1 ≤ p0 ≤ q − 1.
So far we have seen that the dimensions of times τm,n coincide with the dimensions of
the coupling constants λm,n. However, as it was mentioned in the section 2.3, they do not
necessarily coincide but can have a non-linear relation like in (2.22) τm,n= Cm,nµδm,n+ λm,n+ X m1,n1 X m2,n2 C(m1n1)(m2n2) m,n λm1n1λm2n2+ + X m1,n1 X m2,n2 X m3,n3 C(m1n1)(m2n2)(m3n3) m,n λm1n1λm2n2λm3n3+ . . . . (3.20)
There is a strong restriction on the possible terms, namely, they must have the same scaling dimension as λm,n. These relations for the series of (2, 2s+1) Minimal Gravities were found
in [2]. We are aiming to find them in the case of (3, 3s + p0) Minimal Gravity. We will
do it after introducing an explicit expression for the partition function which satisfies the equation (3.4).
3.2 Free energy in the Douglas approach
The free energy F in the Douglas approach satisfies the differential equation ∂2F
∂x2 = u ∗
1 (3.21)
where u∗
1 is the solution of Douglas string equation (3.5) or, equivalently, (3.8).
Nevertheless it is desirable to have an explicit expression for the partition function. Such an expression exists due to relation of the partition function satisfying the equa-tion (3.21) to the Sato’s tau function [24]. Let us introduce necessary definitions to write
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down the explicit formula for the partition function satisfying the equation (3.21). (This is deeply related to Frobenius manifolds. More details can be found in appendix A.)
First of all introduce an algebra of polynomials modulo the polynomial Q′, where prime
denotes the derivative over p
A = C[p]/Q′. (3.22)
Also define a bilinear form on the space of such polynomials (P1(p), P2(p)) := Res
P1(p)P2(p)
Q′
, (3.23)
where, as before, the residue is taken at p = ∞. Define multiplication on the space A by introducing some basis φα and define
φαφβ = Cαβγ φγ mod (Q′). (3.24)
Besides we always take the unity as the first element of the basis: φ1 = 1. Thus we have
Cα 1β = δαβ.
One then finds that
Resφαφβφγ Q′ = C δ αβ · Res φδφγ Q′ = C δ αβgδγ= Cαβγ (3.25)
where the indices are raised and lowered by the metric gαβ
gαβ = (φα, φβ). (3.26)
There are of course many possible choices for the basis φα. However there are two
particularly useful for us. The first one is just to take as a basis the monomials pα.
Besides, there is another useful basis, namely φα= ∂Q ∂vα (3.27) where vα= − q q − αResQ q−α q . (3.28)
This basis has a nice property that the metric in this basis takes a form gαβ = δα+β,q (see
appendix A). Using (3.22)–(3.26) one can prove that in this basis [25] Cαβγ =
∂3F
∂vα∂vβ∂vγ. (3.29)
where the structure constants Cαβγ are evaluated in the basis ∂v∂Qα. The proof of this relation can be found in [25] and also in the appendix A. The formulae (3.29) together with (3.22)–(3.26) gives the structure of the Frobenius manifold on A [22], vα being the
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The claim is that the free energy can be written as follows F = 12 Z u∗ 0 Cαβγ ∂S ∂uβ ∂S ∂uγdu α (3.30)
where u∗ is an appropriate solution of Douglas string equation or equivalently an
appro-priate critical point of the action S. Here the structure constants Cαβγ are evaluated in the
basis φα= pα. The indices are taken up or down by the metric gαβ.
To make sure that this formula for the partition function makes sense, we need to verify two points.
The first is that the free energy F does not depend on the contour of integration. Equivalently, the differential one-form Ω = Cαβγ∂v∂Sβ∂v∂Sγdvα must be closed. It is convenient to use the flat coordinates vα. This is checked by direct calculation of de Rham differential of Ω and using two properties, namely, associativity of the algebra A and the recursion relation for hamiltonians Sk,α
Cαβγ Cγδφ = Cαγφ Cβδγ , (3.31) ∂2θ α,n ∂vβ∂vγ = C δ βγ ∂θα,n−1 ∂vδ (3.32)
where θα,n is the same function as Sn,α up to a normalization factor which is defined in
the appendix A. The proof of the recursion relation is also provided in the appendix A. The second is that the partition function (3.30) must satisfy the equation (3.21). This is easily verified by direct differentiation of (3.30) and using that t0,1 = x and Cα1β = δ
β α.
Now we have a nice explicit expression (3.30) for the partition function which is conve-nient to use. In order to calculate the correlation numbers we make the substitution (3.20) in the partition function (3.30) and then just take derivatives
Zm1n1...mNnN = ∂ ∂λm1n1 . . . ∂ ∂λmNnN λm,n=0 for (m,n)6=(1,1) F[t(λ)]. (3.33)
where after taking derivatives we take λm,n = 0 except for the cosmological constant
µ = λ1,1.
Now we are going to consider explicitly examples of (2, 2s + 1) and (3, 3s + p0) Minimal
Gravity. The former was considered in [2], however we rederive the results of that paper with a few details changed. This will allow us to make some generalizations to the case (3, 3s + p0).
4 (q, p) = (2, 2s + 1) Minimal Gravity
In this case we have the polynomial Q (3.6) and the free energy (3.30)
Q = p2+ u, F = 1
2 Z u∗
0
Su2(u)du (4.1)
where u∗ is the solution of the equation
Su≡ ∂S ∂u = u s+1+ s X n=1 τ1,nus−n= 0 (4.2)
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where for simplicity we have changed the normalization of the times τm,n and overall
normalization of the action (3.18).
To get the generating function for the correlation numbers one needs the resonance relations5 τ1,k+1= λk+ Ckµδk+ X l,k1 Clk1 k µ l λk1+ X l,k1,k2 Clk1k2 k µ l λk1λk2+ . . . X l,k1,...,kn Clk1...kn k µ l λk1. . . λkn+ . . .(4.3)
where all the powers of µ should appear with non-negative integer powers. In Liouville Minimal Gravity these relations always contain only finite number of terms. This due to the restriction that all the terms must be of the same gravitational dimension.
Now we can insert the resonance relations (4.3) into the partition function and polynomial Su. The result is of the form
F = Z0+ s−1 X k=1 λkZk+ 1 2 s−1 X k1,k2=1 λk1λk2Zk1k2+ . . . (4.4) Su = Su0+ s−1 X k=1 λkSuk+ 1 2 s−1 X k1,k2=1 λk1λk2S k1k2 u + . . . (4.5)
Also from the original form of the polynomial Su one finds that
Su0(u) = us+1+ Bµus−1+ Cµ2us−3+ . . . (4.6) Suk(u) = Akus−k−1+ Bkµus−k−3+ Ckµ2us−k−5+ . . . (4.7) Sk1k2 u (u) = Ak1k2u s−k1−k2−3+ B k1k2µu s−k1−k2−5+ C k1k2µ 2us−k1−k2−k3−7+ . . . (4.8) . . .
where all the polynomials have certain parity since µ ∼ u2.
From the discussion of section 3.1 we find that the dimensions in this case are λk∼ µ k+2 2 (4.9) F ∼ µ2s+32 (4.10) Zk1...kn ∼ µ 2s+3−P(ki+2) 2 (4.11)
As usually in the spirit of the scaling theory of criticality, we are interested only in the singular part of the partition function and disregard the regular part as non-universal.
Notice that Zk1...kn is always singular if P ki is even. On the other hand when P ki is odd and additionally
n
X
i=1
ki ≤ 2s + 3 − 2n (4.12)
the correlation number Zk1...kn involves only non-negative integer powers of µ and thus is non-singular. This inequality always holds for one- and two-point correlation numbers. So we shall consider the sector of oddP ki only starting from the three point correlation
numbers.
5It is convenient to make a shift of indices here in order to single out the cosmological constant µ since
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4.1 Dimensionless setup
As in [2] it is convenient to switch to dimensionless quantities sk= gk g u −(k+2) 0 λk (4.13) Su(u) = gus+10 Yu(u/u0) (4.14) F = g2u2s+30 Z (4.15) where u0= u∗(λ = 0) ∼ µ 1 2, gk= (p−k−1)! (2p−2k−3)!! and g = (p+1)! (2p+1)!!.6
Then one has
Z = 12 Z x∗
0
Yu2(x)dx (4.16)
where x = u
u0 and x∗ = x∗(s) is an appropriate zero of the polynomial Yu(x). Notice that x∗(s = 0) = 1.
Similarly to dimensional quantities one has expansions
Z = Z0+ s−1 X k=1 skZk+ 1 2 s−1 X k1,k2=1 sk1sk2Zk1k2+ . . . (4.17) Yu = Yu0+ s−1 X k=1 skYuk+ 1 2 s−1 X k1,k2=1 sk1sk2Y k1k2 u + . . . (4.18) Yu0(x) = C0xs+1+ C0′xs−1+ . . . (4.19) Yuk(x) = Ckxs−k−1+ Ck′xs−k−3+ . . . (4.20) Yk1k2 u (x) = Ck1k2x s−k1−k2−3+ C′ k1k2x s−k1−k2−5+ . . . (4.21) . . .
4.2 One- and two-point correlation numbers
Using (3.33) and the partition function (4.1) we find one- and two-point correlation numbers Zk= Z 1 0 dxYu0(x)Yuk(x) (4.22) Zk1k2 = Z 1 0 dx(Yk1 u (x)Yuk2(x) + Yu0(x)Yuk1k2(x)) (4.23)
where, due to (4.11), the one-point correlation numbers are singular only for even k and the two-point correlation numbers are singular only for even k1+ k2.
Fusion rules (2.5), (2.6) demand that one-point correlation numbers are zero for k 6= 0
and two point numbers are zero when k1 6= k2. The second term in the two point numbers
is actually absent due to (4.22) and to the fact that the polynomial Yk1k2
u can be written
as a linear combination of Yk u.
6For details see [2]. Our polynomial Y
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Also it is convenient here to introduce a new variable y instead of x y + 1
2 = x
2, dx = dy
2√2(1 + y)12
(4.24) In terms of the variable y the polynomials7 Y0
u, Yuk, . . . will contain all powers instead
of going with step 2 (4.19), (4.20). And the shift was made in order for the interval of integration in (4.22), (4.23) to be [−1, 1] instead of [0, 1].
Thus the fusion rules condition become an orthogonality condition on the polynomials Yu0, Yk u Zk= Z 1 −1 dy 2√2(1 + y)12 Yu0(x)Yuk(x) = 0, k 6= 0 (4.25) Zk1k2 = Z 1 −1 dy 2√2(1 + y)12 Yk1 u (x)Yuk2(x) = 0, k1 6= k2 (4.26)
and, together with the condition Yu0(1) = 0, it determines the polynomials Yu0 and Yk u: s odd Y0 u(y) = P (0,−1 2) s+1 2 (y) − P (0,−1 2) s 1 2 (y) s even Y0 u(y) = x P(0,12) s 2 (y) − P (0,1 2) s−2 2 (y) s+k odd Yk u(y) = P (0,−12) s−k−1 2 (y) s+k even Yk u(y) = xP (0,12) s−k−2 2 (y) where Pn(a,b) is Jacobi polynomial.
Due to the relation between Jacobi polynomials and Legendre polynomials Pn
P(0,−12) n (2x2− 1) = P2n(x) (4.27) xP(0, 1 2) n (2x2− 1) = P2n+1(x) (4.28)
it is of course in agreement with results of the paper [2].
4.3 Three-point correlation numbers
Again using (3.33) and (4.1) we derive the three-point correlation numbers Zk1k2k3 = − Yk1 u Yuk2Yuk3 dY0 u dx x=1 + Z 1 0 dxYk1k2 u Yuk3 = − 1 p + Z 1 0 dxYk1k2 u Yuk3 (4.29)
where we used the properties of Jacobi polynomials to get: Yk
u(x = 1) = 1, dYu0 dx x=1 = 1
2s+1 = 1p. Besides we assume that k1, k2 ≤ k3. Two other integral terms in (4.29) with
permutations of k1, k2, k3 are absent due to (4.22).
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As we will see the first term reproduces the expression from minimal gravity and the role of the second term is to kill the first term when the fusion rules are violated.
Also we need not care about the case of odd k1+ k2+ k3. When k1+ k2+ k3 is odd
and < p the fusion rules are violated but Zk1k2k3 is non-singular (it is an integer positive power of µ). And if k1+ k2+ k3 is odd and ≥ p the integral term is automatically zero.
Thus we focus on the case when k1+ k2+ k3 is even. Fusion rules demand
Z 1 0 dxYk1k2 u Yuk3 = (1 p if k1+ k2 < k3 0 if k1+ k2 ≥ k3 (4.30) Again, switching to variables y (4.24) we get
(s + k1+ k2) odd Yuk1k2(y) = 1p P s−k1−k2−3 2 n=0 (4n + 1)P (0,−1 2) n (y) (s + k1+ k2) even Yuk1k2(y) = xp P s−k1−k2−4 2 n=0 (4n + 3)P (0,1 2) n (y)
At this point it is already reasonable to compare some quantities, which are indepen-dent on the normalization of fields and normalization of correlators, in the Douglas equation approach with those in Minimal Gravity. Namely, one considers the following combination
(Zk1k2k3)2Z0 Q3
i=1Zkiki
(4.31) When the fusion rules for three-point numbers are satisfied Zk1k2k3 = −
1 2s+1 and this quantity gives Q3 i=1(2s − 2ki− 1) (2s + 3)(2s + 1)(2s − 1) (4.32)
which coincides with the value (2.25) from Minimal Gravity if we take (q, p) = (2, 2s + 1), mi = 1, ni = ki+ 1.
4.4 Four-point correlation numbers
Direct calculation gives
Zk1k2k3k4 = − d2Y0 u dx2 (dYu0 dx )3 + P4 i=1 dYuki dx (dYu0 dx )2 − P i<jY kikj u dY0 u dx x=1 + + Z 1 0 dx(Yk1k2 u Yuk3k4+ Yuk1k3Yuk2k4+ Yuk1k4Yuk2k3)+ + Z 1 0 dx(Yk1k2k3 u Yuk4+ Yuk1k2k4Yuk3+ Yuk1k3k4Yuk2 + Yuk2k3k4Yuk1) (4.33)
We assume that k1 ≤ k2 ≤ k3 ≤ k4. The role of the terms in the third line again is to
satisfy the fusion rules by cancelling the terms in the first two lines when the fusion rules are violated. On the other hand the terms in the third line does not appear when the
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fusion rules are satisfied. Thus to evaluate the four point correlation numbers we need only the terms in the first two lines.
First of all it is convenient to change variables from x to y according to (4.24) Zk1k2k3k4 = − 16d2Yu0 dy2 + 4 dY0 u dy (4dYu0 dy )3 + P4 i=14dY ki u dy (4dYu0 dy )2 − P i<jY kikj u 4dYu0 dy y=1 + + Z 1 −1 dy 2√2(1 + y)12 (Yk1k2 u Yuk3k4 + Yuk1k3Yuk2k4 + Yuk1k4Yuk2k3) (4.34)
From the properties of Jacobi polynomials (for both even and odd s, k, k1, k2)
(Yu0)′(1) = 2s + 1 4 , (Y 0 u)′′(1) = (s − 1)(s + 2)(2s + 1)32 , (4.35) (Yuk)′(1) = (s − k − 1)(s − k) 8 , Y k1k2 u (1) = (s − k 1− k2− 1)(s − k1− k2− 2) 2(2s + 1) (4.36)
where prime denotes derivative with respect to y. Using this we evaluate the four point function in the region where the fusion rules are satisfied
Zk1k2k3k4 = 1 2(2s + 1)2 − (s − 1)(s + 2) − 2 + 4 X i=1 F (ki+ 1)− − F (k(12|34)) − F (k(13|24)) − F (k(14|23)) ! (4.37) where F (k) = (s − k − 1)(s − k − 2), k(ij|lm)= min(ki+ kj, kl+ km) (4.38)
A reasonable quantity to evaluate here and compare it with the result from the Minimal Gravity is
(Zk1k2k3k4Z0)2 Q4
i=1Zkiki
(4.39) and using the properties of Jacobi polynomials and assuming that as in (2.27) k1 + k4 ≤ k2 + k3 we get precisely the expression from Minimal Gravity (2.26)
with (q, p) = (2, 2s + 1), mi = 1, ni = ki+ 1.
5 (q, p) = (3, 3s + p0) Minimal Gravity
In this case the polynomial Q and the action S are
Q = p3+ up + v, S(u, v) = Res Qp3+1+ s X n=1 τ1,nQs−n+ p0 3 + p−1 X n=s+1 τ1,nQn−s− p0 3 ! (5.1)
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One takes then an appropriate solution (u∗, v∗) of the string equations
( Su = 0
Sv = 0
(5.2) where the lowered indices u, v denote the derivatives over u and v. Evaluating the structure constants Cβγα we find the free energy (3.30) for this model
F = 1 2 Z γ(λ) Su2−u 3S 2 v du + 2SuSvdv (5.3) where the contour γ(λ) goes from (u, v) = (0, 0) to (u, v) = (u∗(λ), v∗(λ)). The analysis of
the section 3.1 gives
p ∼ µ16, u ∼ µ 1 3, v ∼ µ 1 2 Z ∼ µ1+ p 3, S ∼ µ p+4 6 (5.4)
and the dimensions of the times τ1,k
τ1,k+1∼ µ p+3−|p−3(k+1)| 6 ∼ ( µk+22 , 0 ≤ k ≤ s − 1 µs−k2+p03 , s ≤ k ≤ p − 2 (5.5) 5.1 Resonance relations
As it was discussed earlier, generally the times introduced in the action are not the times corresponding to the perturbations by primary operators in the Minimal gravity. Namely the resonances are possible.
By analysing the dimensions of the times (5.5) we find possible resonances between the times τ1,k+1 in the Douglas equation approach and coupling constants λk in minimal
gravity up to the second order τ1,k+1= λk+ ckµ k+2 2 + s−1 X l=1 (k−l)∈2Z βklµ k−l 2 λl+ X Ck1k2 k λk1λk2+ . . . , 0 ≤ k ≤ s − 1 (5.6) τ1,k+1= λk+ 3s+α−2 X l=k+2 (l−k)∈2Z βklµ l−k 2 λl+ X Ck1k2 k λk1λk2+ . . . , s ≤ k ≤ p − 2 (5.7) Coefficients Ck1k2
k are non-zero only if the resonance condition is satisfied
[λk] = [λk1] + [λk2] (5.8)
for some k, where [λk] is the dimension of the quantity λk. Again by analysing the
dimensions (5.5) we find that Ck1k2 k 6= 0, if 1 ≤ k, k1, k2 ≤ s − 1 or ( 0 ≤ k1≤ s − 1 s ≤ k, k2 ≤ p − 2 (5.9) Once the times τ1,k+1 substituted in (5.3) according to (5.6), (5.7), one evaluates the
n-point correlation number as
Zk1...kn = ∂ ∂λk1 . . . ∂ ∂λkn λ=0 F (5.10)
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5.2 Solutions of the string equations
Now let us describe an appropriate solutions of the string equations (5.2). Of course these equations have a number of different solutions. And it is not an easy task to find them all. However we will show that the equations (5.2) always possess one particular solution with required properties. We are going to use this solution in the partition function (5.3). As a check that we picked out the right solution we will see that it allows to make a correspondence with Minimal Gravity and together with appropriately found resonance relations gives the same results for correlation numbers.8
Since the correlation numbers are determined as derivatives of the partition function taken at λm,n = 0 (except for λ1,1 = µ), it will be crucial to know the solution (u∗, v∗) of
the string equations at such λm,n. We will show that one of the equations (5.2) is always
satisfied by such a v∗ that v∗(λ = 0) = 0. To demonstrate this we will need the parities of
the various parts of the action with respect to the change of the sign of v.
Let us substitute the resonance relations (5.6) and (5.7) into the action (5.1). We get some expression which can be written as an expansion in λk
S(u, v) = S0(u, v) + p−2 X k=1 λkSk(u, v) + 1 2 p−2 X k1,k2=1 λk1λk2S k1k2(u, v) + . . . (5.11)
and dimensional analysis gives
S0 ∼ µp+46 (5.12) Sk1...kn ∼ µp+46 − Pn i=1 p+3−|p−3(ki+1)| 6 (5.13)
Each of the functions Sk1...kn is a polynomial in the variables u, v, µ Sk1...kn = X M,N,K uMvNµK ∼ µp+46 − Pn i=1 p+3−|p−3(ki+1)| 6 (5.14)
The dimensions on the right and left hand sides must be the same M 3 + N 2 + K = p + 4 6 − n X i=1 p + 3 − |p − 3(ki+ 1)| 6 (5.15) It is equivalent to p + n X i=1 ki− N = 2M + 2N + 6K + n X i=1 (p + 3 − |p − 3(ki+ 1)| + ki) (5.16)
One can see that the expression on the right hand side is even. Thus (p +Pn
i=1ki− N) is
also even. Consequently the functions Sk1...kn have definite parities with respect to v
S0(u, −v) = (−1)pS0(u, v) (5.17)
8We checked in particular examples that other solutions do not allow to make correspondence already
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Sk1...kn
(u, −v) = (−1)p+PikiSk1...kn(u, v) (5.18) Thus, depending on the value of p, either ∂S∂u0 or∂S∂v0 is odd function of v. Consequently, v = 0 is always a solution of (5.2) at λ = 0. By u0 we denote the solution for u at λ = 0
S0 v v=0 ≡ 0 S0 u v=0 u=u0 = 0 if p is even (5.19) Su0 v=0 ≡ 0 Sv0 v=0 u=u0 = 0 if p is odd. (5.20)
Also many of the functions among Sk1...kn are odd in v and thus vanish at v = 0. We will extensively use this fact in further considerations.
5.3 Picking out the contour
As it was noticed before, the integral (5.3) does not depend on the contour of integration. So we can take it at our choice. It is convenient to take the contour γ in (5.3) as following
✲ ✻ u v u∗(λ) v∗(λ) ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ γ(λ)
This contour has a good property that it reduces the problem to the polynomials of one variable u. Indeed, in (5.10) we put λ = 0 after taking derivatives. As we discussed above, we pick out the solution of the string equation such that (u∗(λ = 0), v∗(λ = 0)) = (u0, 0).
Thus the contour becomes the line along the u-axis
✲ ✻ u v u0 γ(0) r
So when we evaluate the correlation numbers using (5.10) there will be two kind of terms. The first ones are the non-integral terms, which arise when one of the derivatives act on the upper integration limit in (5.3). In these terms after taking λ = 0 (5.10) the
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functions must be taken at the point (u∗(λ = 0), v∗(λ = 0)) = (u0, 0). The second ones are
the integral terms arising from integrating only the functions under the integral in (5.3). After putting λ = 0 the contour of integration becomes γ(0) and it is going along the u-axis. Thus we must take the functions under the integration at v = 0. So we see that the correlation numbers are determined by the functions Sk1...kn and their derivatives over u, v taken at v = 0.
5.4 Dimensional analysis
The functions Sk1...kn(u, 0) are polynomials in u and µ. It will be important to know their degrees as polynomials in u. Dimensional analysis gives
S0(u, 0) ∼ u3(s+1)2 +1+p02 + . . . (5.21) Sk(u, 0) ∼ u3(s−k−1)2 +1+p02 + . . . , 1 ≤ k ≤ s − 1 u3(k−s+1)2 +1−p02 + . . . , s≤ k ≤ p − 2 (5.22) Sk1k2(u, 0) ∼ u3(s−k1−k2−3)2 +1+p02 + . . . , 1 ≤ k1, k2≤ s − 1 u3(k2−k1−s−1)2 +1−p02 + . . . , 1 ≤ k1≤ s − 1, s≤ k2≤ p − 2 (5.23) Sk1k2k3(u, 0) ∼ u3(s−k1−k2−k3−5)2 +1+p02 + . . . , 1 ≤ k1, k2, k3≤ s − 1 u3(k3−k1−k2−s−3)2 +1−p02 + . . . , 1 ≤ k1, k2≤ s − 1, s≤ k3≤ p − 2 u3(k2+k3−k1−3s−1)2 +1−3p02 + . . . , 1 ≤ k1≤ s − 1, s≤ k2, k3≤ p − 2 u3(k1+k2+k3−5s+1)2 +1−5p02 + . . . , s≤ k1, k2, k3≤ p − 2 (5.24) Sk1k2k3k4(u, 0) ∼ u3(s−k1−k2−k3−k4−7)2 +1+p02 + . . . , 1 ≤ k1, k2, k3, k4≤ s − 1 u3(k4−k1−k2−k3−s−5)2 +1−p02 + . . . , 1 ≤ k1, k2, k3≤ s − 1, s≤ k4≤ p−2 u3(k3+k4−k1−k2−3s−3)2 +1−3p02 + . . . , 1 ≤ k1, k2 ≤ s − 1, s≤ k3, k4≤ p−2 u3(k2+k3+k4−k1−5s−1)2 +1−5p02 + . . . , 1 ≤ k1 ≤ s − 1, s≤ k2, k3, k4≤ p−2 u3(k1+k2+k3+k4−7s+1)2 +1−7p02 + . . . , s≤ k1, k2, k3, k4≤ p − 2 (5.25)
where the dots represent the terms involving cosmological constant µ. Since µ ∼ u3 the
degrees of u in each polynomial go with the step 3. Also it is worth noticing that if the degree of the polynomial Sk1...kn is non-integer or negative, it means that it is actually zero. This can be seen from the expression for the action (5.1). Indeed, it would just mean that the corresponding residue vanishes.
Also we disregard all the correlation numbers which are proportional to the integer powers of the cosmological constant µ as non-universal. So it is important to know the dimensions of the correlation numbers in order to check whether we have a non-trivial fusion rule for them
hOki ∼ ( µs−k2+p03 , 1 ≤ k ≤ s − 1 µk+22 , s≤ k ≤ p − 2 (5.26) hOk1Ok2i ∼ µs−k1+k22 +p03−1, 1 ≤ k1, k2≤ s − 1 µk2−k12 , 1 ≤ k1≤ s − 1, s≤ k2≤ p − 2 µk1+k22 −s−p03+1, s≤ k1, k2≤ p − 2 (5.27)
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hOk1Ok2Ok3i ∼ µs+p03−k1+k2+k3+42 , 1 ≤ k1, k2, k3≤ s − 1 µk3−k1−k2−22 , 1 ≤ k1, k2≤ s − 1, s≤ k3≤ p − 2 µk2+k3−k12 −s−p03 , 1 ≤ k1≤ s − 1, s≤ k2, k3≤ p − 2 µk1+k2+k3+22 −2s−2p03 , s≤ k1, k2, k3≤ p − 2 (5.28) hOk1Ok2Ok3Ok4i ∼ µs+1+α3−k1+k2+k3+k4+82 , 1 ≤ k1, k2, k3, k4≤ s − 1, µk4−k1−k2−k3−42 , 1 ≤ k1, k2, k3≤ s − 1, s≤ k4≤ p − 2, µ−s+1−p03+k3+k4−k1−k2−42 , 1 ≤ k1, k2≤ s − 1, s≤ k3, k4≤ p − 2, µ−2s+1−2p03 +k2+k3+k4−k1−22 , 1 ≤ k1≤ s1, s≤ k2, k3, k4≤ p − 2, µ−3s+1−p0+k1+k2+k3+k42 , s≤ k1, k2, k3, k4≤ p − 2 (5.29)Now we are going to evaluate zero-, one-, two, three- and four-point correlation numbers.
5.5 Zero-, one- and two-point numbers
Before going into details let us point out a few issues.
First of all note that, when evaluating one- and two-point numbers, we do not need to differentiate over the limits of integration since at (u∗, v∗)|λ=0 these terms give zero
because of the string equation.
Next, when taking λ = 0 after differentiating over λ’s in (5.10), one finds that many of the functions Sk1...kn
u , Svk1...kn are odd in v due to (5.17)–(5.18) and thus vanish at v = 0 .
Taking these things into account one evaluates zero-, one- and two-point correlation numbers. The result is summarized in the table
p even p - odd Z0= 12R0u0(Su0)2du Z0 = −16R0u0(Sv0)2udu Zk= Ru0 0 S0uSukdu Zk= −13 Ru0 0 Sv0Svkudu k1, k2 even Zk1k2= Ru0 0 (Suk1Suk2+Su0Suk1k2)du Zk1k2= − 1 3 Ru0 0 (Svk1Svk2+Sv0Svk1k2)udu k1, k2 odd Zk1k2= Ru0 0 (Su0Suk1k2−u3Svk1Svk2)du Zk1k2= Ru0 0 (S k1 u Suk2−u3Sv0Svk1k2)du k1+ k2 odd Zk1k2 = 0 Zk1k2 = 0
where all the polynomials are taken at v = 0 since the contour of integration goes along the u-axis.
Now for both p - even or p - odd the analysis is similar to the case of Lee-Yang series (2, 2s + 1). Similarly to that case we switch from S(u, 0) and u to dimensionless quantities Y (x) and x = uu
0 respectively. We do not need the precise relation between dimensional action S and dimensionless Q and do not write it down in order not to confuse the reader
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with unnecessary messy coefficients. Also we implicitly switched from λk to dimensionless
quantities sk and we now assume that ∂k= ∂s∂
k.
Also as in the case of Lee-Yang series we use the variable y y + 1 2 = x 3, dx = dy 3√3 2(1 + y)23 (5.30) Following the logic similar to Lee-Yang series we receive
p even Yu0 = x2(p0−1)(P(0, 2 3(2p0−3)) s−p0+2 2 (y) − P (0,23(2p0−3)) s−p0 2 (y)) p odd Yv0= x2−p0(P (0,23(1−p0)) s+p0−1 2 (y) − P (0,23(1−p0)) s+p0−3 2 (y)) (p + k) even, k < s Yk u = x2(p0−1)P (0,2 3(2p0−3)) s−k−p0 2 (y) (p + k) odd, k < s Yvk= x2−p0P (0,2 3(1−p0)) s−k+p0−3 2 (y) (p + k) even, k ≥ s Yk u = x2(2−p0)P (0,23(3−2p0)) k−s+p0−2 2 (y) (p + k) odd, k ≥ s Yk v = xp0−1P (0,23(p0−2)) k−s−p0+1 2 (y) where all the polynomials are evaluated at v = 0.
5.6 Three-point correlation numbers
For the three-point correlation numbers a direct calculation gives Zk1k2k3 = Sk1 u Sku2 − u0 3 S k1 v Svk2 ∂k3u∗+ S k1 u Svk2∂k3v∗+ (5.31) + Z u0 0 du Sk1 u Suk2k3− u 3S k1 v Svk2k3 + Z u0 0 du Su0Sk1k2k3 u − u 3S 0 vSvk1k2k3 +permutations Taking derivative of the equations (5.2) one gets at sk= 0
Suk+ Suu0 ∂ku∗+ Suv0 ∂kv∗ = 0 (5.32)
Svk+ Svu0 ∂ku∗+ Svv0 ∂kv∗ = 0 (5.33)
So using the parities of polynomials Sk1...kn one has at v = 0
p even p odd k even ∂ku∗ = −S k u S0 uu ∂ku∗ = − Sk v S0 uv k odd ∂kv∗ = −S k v S0 vv ∂kv∗ = − Sk u S0 uv
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where for each case only non-zero derivative is represented. For instance for even p and k we have ∂kv∗ = 0.
Now using this expressions for derivatives of (u∗, v∗) in three-point numbers and, again,
using the parities of Sk1...kn one gets the following result for Z
k1k2k3 at v = 0 p even p odd k1, k2, k3 even −S k1 u Suk2Sk3u S0 uu +R S 0 uSuk1k2k3 u30S k1 v Svk2Sk3v S0 uv − R u 3S 0 vSvk1k2k3− +R (Sk1 u Suk2k3+ permutations) − R u 3(S k1 v Skv2k3+ permutations) k1, k2 even k3 odd 0 0 k1 even u30S k1 u Svk2Sk3v S0 uu − Sk1u Svk2Sk3v S0 vv +R S 0 uS k1k2k3 u + − Svk1Sk2u Suk3 S0 uv − R u 3S 0 vS k1k2k3 v + k2, k3 odd +R (Suk1Suk2k3−u3(Svk2Skv1k3+ Svk3Skv1k2)) +R (Suk2Sku1k3+ Suk3Suk1k2−u3Svk1Svk2k3) k1, k2, k3 odd 0 0
where the off-integral terms are evaluated at u = u0 and all the integrals are taken from
0 to u0.
The simplest case to analyse here is when p is even and all k are even, that is, in the upper left column of the tableau. So let us concentrate on this case
Zk1k2k3= − Sk1 u Suk2Suk3 S0 uu u=u0 + Z u0 0 Su0Sk1k2k3 u du+ Z u0 0 (Sk1 u Suk2k3+Suk2Suk1k3+Suk3Suk1k2)du. (5.34)
• 1 ≤ k1, k2, k3 ≤ s −1; p, k — even. In this case the 3-point correlation numbers are
singular due to eq. (5.28). The expression for them reduces to (we assume that k3 > k1, k2)
Zk1k2k3 = − Yk1 u Yuk2Yuk3 (Y0 u)′ x=1+ Z 1 0 Yk3 u Yuk1k2dx (5.35)
where we switched to dimensionless quantities and prime denotes the derivative over x. Two other integral terms are absent because the degree of the polynomial Sk2k3
u is less
than that of the Jacobi polynomial Sk
u. The other term is treated similarly. Thus to satisfy
fusion rules (2.7) one needs the following condition to hold Z 1 0 Yk3 u Yuk1k2dx = ( 0, if k3 ≤ k1+ k2 1 p, if k3 > k1+ k2 (5.36) This determines Yk1k2 u = 1 p s−k1−k2−p0−2 2 X k=0 (6k+4p0−3)x2(p0−1)P (0,23(2p0−3)) k , 1 ≤k1, k2≤s−1 (5.37)
Now we can evaluate the correlation numbers when the fusion rules are satisfied Z0 =
p
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Zkk= 1 p−3(k+1), 1 ≤ k ≤ s − 1, k even 1 3(k+1)−p, s ≤ k ≤ p − 2, k even (5.39) Zk1k2k3 = − 1 p, 1 ≤ ki ≤ p − 2, k even. (5.40)The quantity that doesn’t depend on the normalization of the operators and correlators is (Zk1k2k3) 2Z 0 Q3 i=1Zkiki = Q3 i=1|p − kiq| p(p + q)(p − q) (5.41)
where p = 3s + p0, q = 3. It is in agreement with the direct calculation of the integrals
over the moduli space in Minimal Gravity (2.25).
For the other ranges of parameters k1, k2, k3 we will see that the three-point correlation
numbers are always given by (5.40) when the fusion rules are satisfied. Thus, the result for those cases also coincides with the direct calculations in Liouville Minimal Gravity. • 1 ≤ k1, k2 ≤ s −1, s ≤ k3 ≤ p −2; p, ki — even. In this case k1+ k2≤ 2(s − 2) ≤
p − 2. Thus the fusion rules (2.7) are violated if k3> k1+ k2, or since all ki are even, it is
equivalent to k3 ≥ k1+k2+2. Thus the 3-point correlation numbers are non-singular (5.28)
when the fusion rules are violated.
On the other hand if the fusion rules are satisfied then the 3-point correlation numbers are singular and the expression for them reduces to
Zk1k2k3 = − Yk1 u Yuk2Yuk3 (Y0 u)′ x=1+ Z 1 0 Yu0Yk1k2k3 u dx + Z 1 0 (Yk1 u Yuk2k3+ Yuk2Yuk1k3+ Yuk3Yuk1k2)dx. (5.42) The integral term here is actually zero, when the fusion rules are satisfied, i.e. when k3 ≤ k1 + k2. Indeed, consider the term
R1
0 Yu0Yuk1k2k3dx. When k3 ≤ k1 + k2 the
degree (5.24) of the polynomial Yk1k2k3
u is less than zero,
3(k3−k1−k2−s−3) 2 + −1−p0 2 ≤ −3(s+3)2 + −1−p0
2 < 0. This actually means that the polynomial vanishes in this case
because the functions Yk1...kn(x) were defined as coefficients of the expansion in λ
k of
the polynomial Y (x). So the functions Yk1...kn(x) must be polynomials. Similarly, when k3 ≤ k1+ k2, the polynomials Yuk1k2, Yuk2k3, Yuk1k3 vanish. Thus we arrive at the following
expression for the three-point correlation numbers when the fusion rules are satisfied Zk1k2k3 = − Yk1 u Yuk2Yuk3 (Y0 u)′ x=1= − 1 p (5.43)
• 1 ≤ k1 ≤ s −1, s ≤ k2, k3 ≤ p −2; p, ki — even. In this case the 3-point numbers
are once again singular (5.28). The expression for them is Zk1k2k3 = − Yk1 u Yuk2Yuk3 (Y0 u)′ x=1+ Z 1 0 Yk2 u Yuk1k3dx. (5.44)
If additionally k1 + k2 ≤ p − 2 then the fusion rules (2.7) require that the polynomials
Yk1k3 u satisfy Z 1 0 Yk2 u Yuk1k3dx = ( 0, if k3 ≤ k1+ k2 1 p, if k3 > k1+ k2 , k1+ k2≤ p − 2. (5.45)
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If k1+ k2> p − 2 then Z 1 0 Yk2 u Yuk1k3dx = ( 0, if k1+ k2+ k3≤ 2(p − 2) 1 p, if k1+ k2+ k3 > 2(p − 2) , k1+ k2> p − 2. (5.46)However, notice that, when k3 ≤ k1+k2 the integral
R1
0 Yuk2Yuk1k3dx automatically vanishes
since the degree of the polynomial Yk1k3
u is less then the degree of the orthogonal Jacobi
polynomial Yk2
u . Thus we can’t satisfy the condition (5.46), since k1+ k2 > p − 2 ≥ k3.
This is the first time where we encounter problems with satisfying the selection rules. Nevertheless, despite the fact that we can not satisfy all of the selection rules, we will see that after satisfying a part of the selection rules we can get some sensible results. For the three-point correlation numbers this will allow us to find all the polynomials Yk1k2
u .
As a check that we get the right expressions for these polynomials we will calculate the four-point correlation numbers which satisfy the fusion rules (and thus are not zero) and involve the polynomials Yk1k2
u . We will see that they coincide with four-point correlation
numbers from Minimal Liouville Gravity (2.26).
Thus the only non-trivial condition which we have for the polynomials Yk1k3
u is (5.45).
This determines the polynomials Yk1k3
u . Yk1k3 u = 1 p k3−k1−s+p0−4 2 X k=0 (6k + 9 − 4p0)x2(2−p0)P (0,23(3−2p0)) k , ( 1 ≤ k1 ≤ s − 1, s ≤ k3≤ p − 2 (5.47) • s ≤ k1, k2, k3 ≤ p −2; p, ki are even. In this case the three-point correlation
numbers are singular (5.28) and the expression for them reduces to Zk1k2k3 = − Yk1 u Yuk2Yuk3 (Y0 u)′ x=1+ Z 1 0 Yu0Yk1k2k3 u dx. (5.48)
Thus to satisfy the conformal and selection rules we need Z 1 0 Yu0Yk1k2k3 u dx = ( 0, if k3≤ k1+ k2 1 p, if k3 > k1+ k2 , for k1+ k2 ≤ p − 2 (5.49) and Z 1 0 Yu0Yuk1k2k3dx = ( 0, if k1+ k2+ k3≤ 2(p − 2) 1 p, if k1+ k2+ k3> 2(p − 2) , for k1+ k2 > p − 2. (5.50)
Observe that the integralR1
0 Yu0Yuk1k2k3dx vanishes automatically when k1+ k2+ k3≤
2(p − 2) for the same reason as did the integralR1
0 Yuk2Yuk1k3dx in (5.46). Thus we see that
if k1+ k2≤ p − 2 then k1+ k2+ k3≤ 2(p − 2) and the condition (5.49) cannot be satisfied
because the integralR1
0 Yu0Yuk1k2k3dx vanishes automatically.
This finishes the analysis of the three-point correlation numbers and we shall move on to the four-point case now.
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5.7 Four-point correlation numbers
Direct calculation gives for even p, ki
Zk1k2k3k4 = Z (0) k1k2k3k4 + Z (I) k1k2k3k4 (5.51) where Zk(0)1k2k3k4= − (Y 0 u)′′ ((Y0 u)′)3 + P4 i=1(Yuki)′ ((Y0 u)′)2 − P i<jY kikj u (Y0 u)′ ! x=1 + + Z 1 0 dx(Yk1k2 u Yuk3k4 + Yuk1k3Yuk2k4+ Yuk1k4Yuk2k3) (5.52) Zk(I) 1k2k3k4= Z 1 0 dx(Yk1k2k3 u Yuk4+Yuk1k2k4Yuk3+Yuk1k3k4Yuk2+Yuk2k3k4Yuk1+Yu0Yuk1k2k3k4) (5.53) where prime denotes the x-derivative. We assume that k1 ≤ k2 ≤ k3 ≤ k4.
• 1 ≤ k1, k2, k3, k4 ≤ s −1; p, ki — even. First of all the dimensional analysis
gives that in this case the 4-point correlation numbers are singular (5.29). Polynomials involved in Zk(0)
1k2k3k4 are known from the previous consideration and polynomials involved in Zk(I)
1k2k3k4 are not known yet.
Since k1≤ k2 ≤ k3≤ k4, the orthogonality of Jacobi polynomials gives
Zk(I)1k2k3k4 = Z 1 0 dxYk1k2k3 u Yuk4 (5.54) Since Yk4
u is the Jacobi polynomial, Z (I)
k1k2k3k4 automatically equals zero when k4 ≤ k1+ k2+ k3+ 2.
The fusion rules (2.8) in this case are
Zk1k2k3k4 = 0, iff k4> k1+ k2+ k3 (5.55) Thus we need Z 1 0 dxYk1k2k3 u Yuk4 = ( 0, k4 ≤ k1+ k2+ k3 −Zk(0)1k2k3k4, k4 > k1+ k2+ k3 (5.56)
Using the properties of Jacobi polynomials (see appendix B) to evaluate Zk(0)1k2k3k4 we find that Zk(0)1k2k3k4= ( 1 4p2(2+k1+k2+k3−k4)(2p−3Pi=14 (ki+1)), if k4> k1+k2+k3 1 2p2(1 + k1)(2p − 3 P4 i=1(ki+ 1)), if k1+ k4 ≤ k2+ k3 (5.57) Thus we find from the fusion rule (5.56)
Yk1k2k3 u = 1 p2 s−k1−k2−k3−p0−4 2 X k=0 ckx2(p0−1)P (0,23(2p0−3)) k (y), 1 ≤ k1, k2, k3 ≤ s − 1 (5.58)
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where ck= 1 4(6k + 4p0− 3) 2k + 3 X i=1 ki− s + p0+ 2 2p − 3 3 X i=1 ki+ 3(2k − s + p0) − 12 (5.59) Also, when k1+ k4≤ k2+ k3 and the fusion rules are satisfied we find thatZk1k2k3k4Z0 Q4 i=1Zkiki 12 = Q4 i=1|p − 3(ki+ 1)| 1 2 2p(p + 3)(p − 3) (1 + k1) 2p − 3 4 X i=1 (ki+ 1) (5.60)
exactly coincides with the result of direct calculation [5] in Minimal Gravity (2.26), where mi = 1, ni= ki+ 1, q = 3.
So far the calculations of four-point correlation numbers are in full agreement with the Liouville Minimal Gravity. However, in the next cases we will have only partial agreement. • 1 ≤ k1, k2, k3 ≤ s −1, s ≤ k4 ≤ p −2; p, ki — even. The dimensional
anal-ysis (5.29) gives that in this case if k4 < k1 + k2 + k3 + 4 then the 4-point numbers are
singular. Meanwhile the selection rules (2.8) are violated if k4 > k1+ k2+ k3. Thus there is
a “window” at k4= k1+ k2+ k3+ 2 when it needs to be checked that the 4-point numbers
are zero. So one needs to verify that Zk(0)1k2k3k4+Zk(I)1k2k3k4 = 0 when k4= k1+k2+k3+2.
Be-sides, when k4 = k1+k2+k3+2 there are no counter terms Zk(I)1k2k3k4 involving polynomials Yk1k2k3
u . The expression for the four-point correlation numbers reduce to
Zk1k2k3k4 = − (Yu0)′′ ((Y0 u)′)3 + P4 i=1(Yuki)′ ((Y0 u)′)2 − P i<jY kikj u (Y0 u)′ ! x=1 + (5.61) + Z 1 0 dx(Yk1k2 u Yuk3k4+ Yuk1k3Yuk2k4 + Yuk1k4Yuk2k3)
All the polynomials in this expression are known from the previous discussion. However we checked that this expression does not give zero when k4= k1+ k2+ k3+ 2.
• 1 ≤ k1, k2 ≤ s −1, s ≤ k3, k4 ≤ p −2; p, ki — even. Dimensional analysis (5.29)
gives that the 4-point correlation numbers are singular. The expression (5.51) reduces to
Zk1k2k3k4 = Z (0) k1k2k3k4 + Z 1 0 dxYk1k2k4 u Yuk3 (5.62)
Thus to satisfy the fusion rules (2.8) we need Z 1 0 dxYk1k2k4 u Yuk3 = ( 0, k4 ≤ k1+ k2+ k3 −Zk(0)1k2k3k4, k4 > k1+ k2+ k3 , when k1+ k2+ k3 ≤ p − 2 (5.63) and Z 1 0 dxYk1k2k4 u Yuk3 = ( 0, k1+ k2+ k3+ k4 ≤ 2(p − 2) −Zk(0)1k2k3k4, k1+ k2+ k3+ k4 > 2(p − 2) , k1+k2+k3> p−2. (5.64)
